A Approaches towards quantum gravity

Hawking radiation is a feature of QFT in curved spacetime. It does not require that we quantize gravity - it just requires that we quantize the perturbative fields on the black hole background. In fact we can see very similar physics in at spacetime.

1. Does a QFT in curved spacetime simply replace a Lagrangian in flat Minkowski spacetime with the same Lagrangian in a given curved spacetime?

2. Does a QFT in curved spacetime not include the Einstein action?

3. Does a QFT in curved spacetime have limited predictive power than the non-renormalizable quantized Einstein-Hibert action coupled to the same QFT, because we have to choose a specific spacetime metric in the former case in order to draw predictions from the theory?

4. Does a QFT in curved spacetime quantize the matter fields which manifest themselves well below the Planck scale, but use a given classical spacetime metric?

1) Yes, but that is just schematic. What matters is how you solve the quantization procedure, and there are several different approaches that satisfy your criteria (along with several different regularization and renormalization schemes). You should start with reading Birrel and Davies

2) Again it depends on the approach. In the first naive treatment, the Einstein action remains classical (so its variation remains classical). However, in the semiclassical treatment described in Birrel and Davies, the metric tensor is split into a piece that remains classical, and a small perturbation which is 'quantum'. This piece is included in the matter part of the spacetime and the perturbation series is truncated at first order (which avoids the first pure gravity loop divergence at 2 loops). An effective field theory treatment, on the other hand, keeps all the orders, but requires an unbounded amount of high energy experiments to fix the coefficients.

3) The name 'QFT in a curved spacetime' refers to several differerent techniques that may or may not be simply related, so its difficult to answer your question without more specificity. In general yes, you need to specify a background to get results (but this is no different than say asking the question about what happens in a QCD process in an external electromagnetic field, most techniques will make the approximation that the large external electromagnetic field remains classical and will be unnaffected by any tiny backreaction from the process on itself.

how an infinite sum of fractions can add up to an irrational number. Conceptually, something similar happens in Fierz-Pauli theory (self-interacting spin-2 particles); adding an infinite amount of higher order terms results in a background-independent theory. Interactions on a flat spacetime can be added up to give a curved spacetime.